Zero-Error Distributed Compression of Binary Arithmetic Sum

In this paper, we put forward a model of zero-error distributed function compression system of two binary memoryless sources <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"...

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Veröffentlicht in:IEEE transactions on information theory Jg. 70; H. 5; S. 3100 - 3117
Hauptverfasser: Guang, Xuan, Zhang, Ruze
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.05.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Arithmetic
binary arithmetic sum
Channel coding
Coders
Computation
Computational modeling
Decoding
distributed function compression
Errors
Graph coloring
network function computation
Random variables
Source coding
Upper bound
Upper bounds
zero-error compression capacity
ISSN:0018-9448, 1557-9654
Online-Zugang:Volltext
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Zusammenfassung:In this paper, we put forward a model of zero-error distributed function compression system of two binary memoryless sources <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">Y </tex-math></inline-formula>. In this model, there are two encoders <inline-formula> <tex-math notation="LaTeX">\mathbf {En1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathbf {En2} </tex-math></inline-formula> and one decoder <inline-formula> <tex-math notation="LaTeX">\mathbf {De} </tex-math></inline-formula>, connected by two channels <inline-formula> <tex-math notation="LaTeX">(\mathbf {En1}, \mathbf {De}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">(\mathbf {En2}, \mathbf {De}) </tex-math></inline-formula> with the capacity constraints <inline-formula> <tex-math notation="LaTeX">C_{1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">C_{2} </tex-math></inline-formula>, respectively. The encoder <inline-formula> <tex-math notation="LaTeX">\mathbf {En1} </tex-math></inline-formula> can observe <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">(X,Y) </tex-math></inline-formula> and the encoder <inline-formula> <tex-math notation="LaTeX">\mathbf {En2} </tex-math></inline-formula> can observe <inline-formula> <tex-math notation="LaTeX">Y </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">(X,Y) </tex-math></inline-formula> according to the two switches <inline-formula> <tex-math notation="LaTeX">{\mathbf {s}}_{1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">{\mathbf {s}}_{2} </tex-math></inline-formula> open or closed (corresponding to taking values 0 or 1). The decoder <inline-formula> <tex-math notation="LaTeX">\mathbf {De} </tex-math></inline-formula> is required to compress the binary arithmetic sum <inline-formula> <tex-math notation="LaTeX">f(X,Y)=X+Y </tex-math></inline-formula> with zero error by using the system multiple times. We use <inline-formula> <tex-math notation="LaTeX">({\mathbf {s}}_{1}{\mathbf {s}}_{2};C_{1}, C_{2}; f) </tex-math></inline-formula> to denote the model in which it is assumed that <inline-formula> <tex-math notation="LaTeX">C_{1}\geq C_{2} </tex-math></inline-formula> by symmetry. The compression capacity for the model is defined as the maximum average number of times that the function <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> can be compressed with zero error for one use of the system, which measures the efficiency of using the system. We fully characterize the compression capacities for all the four cases of the model <inline-formula> <tex-math notation="LaTeX">({\mathbf {s}}_{1}{\mathbf {s}}_{2};C_{1}, C_{2}; f) </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">{\mathbf {s}}_{1}{\mathbf {s}}_{2}=00,01,10,11 </tex-math></inline-formula>. Here, the characterization of the compression capacity for the case <inline-formula> <tex-math notation="LaTeX">(01;C_{1},C_{2};f) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">C_{1}>C_{2} </tex-math></inline-formula> is highly nontrivial, where a novel graph coloring approach is developed. Furthermore, we apply the compression capacity for <inline-formula> <tex-math notation="LaTeX">(01;C_{1},C_{2};f) </tex-math></inline-formula> to an open problem in network function computation that whether the best known upper bound of Guang et al. on computing capacity is in general tight. Up to now, we are not aware of any example for which this upper bound is not tight. By considering a network function computation model transformed from <inline-formula> <tex-math notation="LaTeX">(01;C_{1},C_{2};f) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">C_{1}>C_{2} </tex-math></inline-formula>, we give the answer that in general the upper bound of Guang et al. is not tight.
Bibliographie:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3319976